09-240/Classnotes for Tuesday September 29

09-240/Navigation Panel   Additions to the MAT 240 web site no longer count towards good deed points # Week of... Notes and Links 1 Sep 7 Tue, About, Thu 2 Sep 14 Tue, HW1, HW1 Solution, Thu 3 Sep 21 Tue, HW2, HW2 Solution, Thu, Photo 4 Sep 28 Tue, HW3, HW3 Solution, Thu 5 Oct 5 Tue, HW4, HW4 Solution, Thu, 6 Oct 12 Tue, Thu 7 Oct 19 Tue, HW5, HW5 Solution, Term Test on Thu 8 Oct 26 Tue, Why LinAlg?, HW6, HW6 Solution, Thu 9 Nov 2 Tue, MIT LinAlg, Thu 10 Nov 9 Tue, HW7, HW7 Solution Thu 11 Nov 16 Tue, HW8, HW8 Solution, Thu 12 Nov 23 Tue, HW9, HW9 Solution, Thu 13 Nov 30 Tue, On the final, Thu S Dec 7 Office Hours F Dec 14 Final on Dec 16 To Do List The Algebra Song! Register of Good Deeds Misplaced Material Add your name / see who's in!

WARNING: The notes below, written for students and by students, are provided "as is", with absolutely no warranty. They can not be assumed to be complete, correct, reliable or relevant. If you don't like them, don't read them. It is a bad idea to stop taking your own notes thinking that these notes can be a total replacement - there's nothing like one's own handwriting! Visit this pages' history tab to see who added what and when.

• 

  Yangjiay - Page 1

• 

  Page 2

• 

  Page 3

• 

  Page 4

• 

  Page 5

• 

  Page 6

Vector subspaces

Definition. $\mathbf {W} \subset \mathbf {V}$ is a "subspace" if it is a vector space under the operations it inherits from V.

Theorem. $\mathbf {W} \subset \mathbf {V}$ is a subspace iff it is "closed under addition and vector multiplication by scalars", i.e. $x,y\in \mathbf {W} \Rightarrow x+y\in \mathbf {W}$ and $a\in F,x\in \mathbf {W} \Rightarrow ax\in \mathbf {W}$.

Goal: Every VS has a "basis", so while we don't have to use coordinates, we always can.

Examples of what is not a subspace (without diagrams):

1. A unit circle is not closed under addition of scalar multiplication.
2. The x-axis $\cup$ y-axis is closed under scalar multiplication, but not under addition.
3. A single quadrant of the Cartesian plane is closed under addition, but not under scalar multiplication.

Examples of subspaces:

1. $\{0\}$
2. Any VS (which is a subspace of itself)
3. A line passing through the origin (if it does not pass through the origin, then it is not closed under scalar multiplication)
4. A plane
5. Let $\mathbf {V} =\mathbb {M} _{n\times n}(F)$. If $W=\{A\in \mathbf {V} :A^{\top }=A\}$, then W is a subspace of V. (W is the set of "symmetric" matrices in V; A^T denotes the transpose of A.)
6. $\mathbf {W} =\{A\in \mathbb {M} _{n\times n}:\operatorname {tr} (A)=0\}$

   where $\operatorname {tr} (A)=\sum _{i=1}^{n}a_{ii}$ is the "trace" of A.

   Properties of trace:

   1. $\operatorname {tr} (0\cdot A)=0$
   2. $\operatorname {tr} (cA)=c\cdot \operatorname {tr} (A)$
   3. $\operatorname {tr} (A+B)=\operatorname {tr} (A)+\operatorname {tr} (B)$

   so W is indeed a subspace.

Claim: If W₁ and W₂ are subspaces of V, then

1. $W_{1}\cap W_{2}=\{x\in \mathbf {V} :x\in \mathbf {W} _{1}{\mbox{ and }}\mathbf {W} _{2}\}$ is a subspace of V, W₁, and W₂.
2. But $W_{1}\cup W_{2}=\{x\in \mathbf {V} :x\in \mathbf {W} _{1}{\mbox{ or }}x\in \mathbf {W} _{2}\}$ is a subspace of V iff $\mathbf {W} _{1}\subset \mathbf {W} _{2}$ or $\mathbf {W} _{2}\subset \mathbf {W} _{1}$. (See HW2 pp. 20-21, #19.)

Linear combinations

Definition: A vector u is a "linear combination" (l.c.) of vectors u₁, ..., u_n if there exists scalars a₁, ..., a_n such that

$u=a_{1}u_{1}+a_{2}u_{2}+\ldots +a_{n}u_{n}$

Example: $\mathbb {P} _{n}(F)=\{{\mbox{Polynomials of degree at most }}n{\mbox{ with coefficients in }}F\}$
$=\left\{\sum _{i=0}^{n}a_{i}x^{i}:a_{i}\in F\right\}$

Definition: A subset $S\subset \mathbf {V}$ "generates" or "spans" V iff the set of linear combinations of elements of S is all of V.

Example: Let $\mathbf {V} =\mathbb {M} _{n\times n}(\Re )$
Let $M_{1}={\begin{pmatrix}1&0\\0&0\end{pmatrix}},M_{2}={\begin{pmatrix}0&1\\0&0\end{pmatrix}},M_{3}={\begin{pmatrix}0&0\\1&0\end{pmatrix}},M_{4}={\begin{pmatrix}0&0\\0&1\end{pmatrix}}$
Then $S=\{M_{1},M_{2},M_{3},M_{4}\}$ generates V.

Proof: Given ${\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in \mathbb {M} _{2\times 2}(\Re )$, write
${\begin{pmatrix}a&b\\c&d\end{pmatrix}}=aM_{1}+bM_{2}+cM_{3}+dM_{4}.$

Example: Let $N_{1}={\begin{pmatrix}0&1\\1&1\end{pmatrix}},N_{2}={\begin{pmatrix}1&0\\1&1\end{pmatrix}},N_{3}={\begin{pmatrix}1&1\\0&1\end{pmatrix}},N_{4}={\begin{pmatrix}1&1\\1&0\end{pmatrix}}$
Does $\{N_{1},N_{2},N_{3},N_{4}\}$ generate V?

$M_{1}=-{\frac {2}{3}}N_{1}+{\frac {1}{3}}(N_{2}+N_{3}+N_{4})$

$M_{2}=-{\frac {2}{3}}N_{2}+{\frac {1}{3}}(N_{1}+N_{3}+N_{4})$

$M_{3}=-{\frac {2}{3}}N_{3}+{\frac {1}{3}}(N_{1}+N_{2}+N_{4})$

$M_{4}=-{\frac {2}{3}}N_{4}+{\frac {1}{3}}(N_{1}+N_{2}+N_{3})$

Then ${\begin{pmatrix}a&b\\c&d\end{pmatrix}}=aM_{1}+bM_{2}+cM_{3}+dM_{4}=$

$a\cdot \left(-{\frac {2}{3}}N_{1}+{\frac {1}{3}}(N_{2}+N_{3}+N_{4})\right)+b\cdot \left(-{\frac {2}{3}}N_{2}+{\frac {1}{3}}(N_{1}+N_{3}+N_{4})\right)+\ldots$

Theorem: If $S\in \mathbf {V}$, then $\operatorname {span} (S)=$ {all l.c. of elements of S} is a subspace of V.